caa a22 4500 605496714 CHVBK 20210128100537.0 cr unu---uuuuu 210128e20150401xx s 000 0 eng 10.1007/s10231-013-0384-0 doi (NATIONALLICENCE)springer-10.1007/s10231-013-0384-0 Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem [Elektronische Daten] [Alberto Boscaggin, Fabio Zanolin] We study the second-order nonlinear differential equation $$u'' + a(t) g(u) = 0$$ u ′ ′ + a ( t ) g ( u ) = 0 , where $$g$$ g is a continuously differentiable function of constant sign defined on an open interval $$I\subseteq {\mathbb R}$$ I ⊆ R and $$a(t)$$ a ( t ) is a sign-changing weight function. We look for solutions $$u(t)$$ u ( t ) of the differential equation such that $$u(t)\in I,$$ u ( t ) ∈ I , satisfying the Neumann boundary conditions. Special examples, considered in our model, are the equations with singularity, for $$I = {\mathbb R}^+_0$$ I = R 0 + and $$g(u) \sim - u^{-\sigma },$$ g ( u ) ∼ - u - σ , as well as the case of exponential nonlinearities, for $$I = {\mathbb R}$$ I = R and $$g(u) \sim \exp (u)$$ g ( u ) ∼ exp ( u ) . The proofs are obtained by passing to an equivalent equation of the form $$x'' = f(x)(x')^2 + a(t)$$ x ′ ′ = f ( x ) ( x ′ ) 2 + a ( t ) . Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2013 Boundary value problems nationallicence Indefinite weight nationallicence Necessary and sufficient solvability conditions nationallicence Boscaggin Alberto Department of Mathematics, University of Torino, Via Carlo Alberto 10, 10123, Torino, Italy aut Zanolin Fabio Department of Mathematics and Computer Science, University of Udine, Via delle Scienze 206, 33100, Udine, Italy aut Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/2(2015-04-01), 451-478 0373-3114 194:2<451 2015 194 10231 https://doi.org/10.1007/s10231-013-0384-0 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10231-013-0384-0 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Boscaggin Alberto Department of Mathematics, University of Torino, Via Carlo Alberto 10, 10123, Torino, Italy aut NATIONALLICENCE 700 1- Zanolin Fabio Department of Mathematics and Computer Science, University of Udine, Via delle Scienze 206, 33100, Udine, Italy aut NATIONALLICENCE 773 0- Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/2(2015-04-01), 451-478 0373-3114 194:2<451 2015 194 10231